#72311536

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	#6591514	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄² ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,20)(15,19)(21,32)(22,31)(23,40)(24,33)(25,35)(26,36)(27,37)(28,38)(29,34)(30,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,25)(13,27)(14,29,30,23)(15,24)(17,35)(18,37)(19,33)(20,40,39,34)(21,26,22,28)(31,36,32,38);(1,2)(3,5)(4,6)(7,8)(9,12,15,13)(10,17,19,18)(11,21,24,22)(14,26,30,28)(16,31,33,32)(20,36,39,38)(23,25,29,27)(34,35,40,37);(1,3,4,8)(2,5,6,7)(9,13,15,12)(10,18,19,17)(11,22,24,21)(14,28,30,26)(16,32,33,31)(20,38,39,36)(23,27,29,25)(34,37,40,35);(2,6)(5,7)(10,19)(12,13)(16,33)(20,39)(21,22)(25,27)(26,28)(34,40)

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=fgj,b^a=bh,c^a=ci,c^b=cg,d^a=di,d^b=df,e^a=efgj,e^b=efg,e^c=efgj,e^d=ej,f^a=fi,f^b=fi,g^a=gi,g^b=gi,h^c=hi,h^d=hi,h^f=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311573, #72311804, #72312203
maximal normal subgroup of	#72311573, #72311804, #72312203

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);